Smooth structures on algebraic surfaces with cyclic fundamental group (Q1086889)

The main result is the following theorem: Let \(M_ 0\) and \(M_ 1\) be closed, oriented, smooth 4-manifolds with fundamental group \({\mathbb{Z}}/k\). Suppose that either \(w_ 2(M_ i)\neq 0\) and \(e(M_ i)\geq | \sigma (M_ i)| +4\), or \(w_ 2(M_ i)=0\) and \(e(M_ i)\geq | \sigma (M_ i)| +6\) for \(i=0,1\). Then \(M_ 0\) is homeomorphic to \(M_ 1\), preserving the orientation, if and only if they have the same signature, Euler characteristic and \(w_ 2\)-type. Here \(e(M_ i)\) is the Euler characteristic and \(\sigma (M_ i)\) the signature, and the manifold M has \(w_ 2\)-type (I), (II), or (III) if one of the following holds: (I) \(w_ 2(\tilde M)\neq 0\), (II) \(w_ 2(M)=0\), or (III) \(w_ 2(M)\neq 0\) and \(w_ 2(\tilde M)=0\). This result implies the homeomorphism classification of Dolgachev surfaces: \(D_{p,q}\) and \(D_{p',q'}\) are homeomorphic if and only if \((p,q)=(p',q')=k\) and, when k is even, \((p+q)/k\equiv (p'+q')/k\) (mod 2). Combining this information with results of Donaldson, Lübke and Okonek, and independently Maier, one obtains the following corollaries: A: All the Dolgachev surfaces \(D_{p,q}\) admit infinitely many smooth structures. B: An algebraic surface with a non-trivial finite cyclic fundamental group admits at least two smooth structures.